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3nj-symbols and identities for q-Bessel functions
Groenevelt, Wolter DOI 10.1007/s11139-017-9952-z Publication date 2018 Document Version Final published version Published in
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Groenevelt, W. (2018). 3nj-symbols and identities for q-Bessel functions. The Ramanujan Journal: an international journal devoted to areas of mathematics influenced by Ramanujan, 47(2), 317–337. https://doi.org/10.1007/s11139-017-9952-z
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https://doi.org/10.1007/s11139-017-9952-z
3n j -symbols and identities for q-Bessel functions
Wolter Groenevelt1
Received: 3 February 2017 / Accepted: 12 September 2017 / Published online: 10 November 2017 © The Author(s) 2017. This article is an open access publication
Abstract The 6 j -symbols for representations of the q-deformed algebra of
polyno-mials on SU(2) are given by Jackson’s third q-Bessel functions. This interpretation
leads to several summation identities for the q-Bessel functions. Multivariate q-Bessel functions are defined, which are shown to be limit cases of multivariate Askey–Wilson polynomials. The multivariate q-Bessel functions occur as 3n j -symbols.
Keywords Jackson’s third q-Bessel function · 6 j-symbols · 3nj-symbols ·
Multivariate q-Bessel function· Quantum algebra representations
Mathematics Subject Classification 33D45· 33D50 · 33D80 · 81R50
1 Introduction
It is well known that Wigner’s 6 j -symbols for the SU(2) group are multiples of
hypergeometric orthogonal polynomials called the Racah polynomials. Similarly,
6 j -symbols for the SU(2) quantum group can be expressed in terms of q-Racah
polynomials, which are q-hypergeometric orthogonal polynomials. With this inter-pretation, properties of 6 j -symbols such as summation formulas and orthogonality relations lead to properties of specific families of orthogonal polynomials, see
e.g., [21,22, Chaps. 8, 14].
In this paper, we consider 6 j -symbols for representations of the q-deformed algebra
of polynomials on SU(2). This algebra has as irreducible representations the trivial one,
and a family of infinite-dimensional representations which disappear in the classical
B
Wolter Groenevelt w.g.m.groenevelt@tudelft.nllimit. The 6 j -symbols for tensor products of three infinite-dimensional
representa-tions can be expressed in terms of Jackson’s third q-Bessel funcrepresenta-tions [8]. Note that,
different from the classical 6 j -symbols, these are not polynomials. We consider three
fundamental identities for 6 j -symbols (see e.g., [1]): Racah’s backcoupling identity,
the Biedenharn–Elliott identity and the hexagon identity. These identities are obtained by decomposing 3- or 4-fold tensor product representations in several ways. To keep track of the order of decomposing the representations, it is convenient to identify cer-tain vectors in the representation spaces with binary trees. Then the 6 j -symbols can be considered as coupling coefficients between two of these trees. The identities we obtain can be interpreted as summation identities for q-Bessel functions. We remark that the hexagon identity implies that the q-Bessel functions are matrix elements of an infinite-dimensional solution of the quantum Yang–Baxter equation (or, the star-triangle
equation in IRF-models), see e.g., [10], which should be of independent interest.
We also consider specific 3n j -symbols, which may naturally be considered as mul-tivariate q-Bessel functions. The one variable q-Bessel functions fit into an extended
Askey-scheme [15] of orthogonal q-hypergeometric functions; the original
(q-)Askey-scheme [12] consists of (q-)hypergeometric orthogonal polynomials. We will show
that the multivariate q-Bessel functions fit into an extended Askey-scheme of multivari-ate orthogonal functions of q-hypergeometric type, by showing that the multivarimultivari-ate q-Bessel functions can be obtained as limits of the multivariate Askey–Wilson
poly-nomials defined by Gasper and Rahman [4], which are the q-analogs of Tratnik’s
multivariate Wilson polynomials [19]. The multivariate Askey–Wilson polynomials
can be thought of as being on top of a scheme of multivariate orthogonal polynomials;
several limit cases are considered in [4,5,9]. Geronimo and Iliev [7] obtained
multivari-ate Askey–Wilson functions generalizing the multivarimultivari-ate Askey–Wilson polynomials, which should be on top of the extended Askey-scheme. Several families of orthogonal
polynomials in this scheme and its q = 1 analog are connected to tensor product
representations and binary coupling schemes, see e.g., Van der Jeugt [20], Rosengren
[17], Scarabotti [18], and a recent result [6] by Genest et al.
This paper is organized as follows: In Sect. 2, the quantum algebra Aq(SU(2))
and its representation theory are recalled. In Sect.3, it is shown that the 6 j -symbols
are essentially q-Bessel functions, using a generating function for q-Bessel functions. Using binary trees, we obtain the fundamental identities for 6 j -symbols, leading to
summation formulas for the q-Bessel functions. In Sect.4, we first define multivariate
q-Bessel functions as nontrivial products of q-Bessel functions, and we prove orthog-onality relations. Then we show that these multivariate q-Bessel functions occur as 3n j -symbols, and use this interpretation to find a summation formula.
Notations We use N = {0, 1, 2, . . .} and we use standard notation for
q-hypergeometric functions as in [3].
2 The quantum algebra
A
q(SU(2))
Let q ∈ (0, 1). The q-deformed algebra of polynomials on SU(2) is the complex
unital associative algebraAq= Aq(SU(2)) generated by α, β, γ , δ, which satisfy the
αβ = qβα, αγ = qγ α, βδ = qδβ, γ δ = qδγ,
βγ = γβ, αδ − qβγ = 1 = δα − q−1βγ. (2.1)
Aqis a Hopf-∗-algebra with ∗-structure and comultiplication defined on the
gen-erators by
α∗= δ, β∗= −qγ, γ∗= −q−1β, δ∗= α, (2.2)
(α) = α ⊗ α + β ⊗ γ, (β) = α ⊗ β + β ⊗ δ,
(γ ) = γ ⊗ α + δ ⊗ γ, (δ) = δ ⊗ δ + γ ⊗ β. (2.3)
An irreducible∗-representation of Aqis either 1-dimensional or infinite-dimensional.
The infinite-dimensional irreducible ∗-representations are labeled by φ ∈ [0, 2π),
and we denote a representation byπφ. The representation space ofπφis 2(N). The
generatorsα, β, γ, δ act on the standard orthonormal basis {en | n ∈ N} of 2(N) by
πφ(α) en= 1− q2ne n−1, πφ(β) en= −e−iφqn+1en, πφ(γ ) en= eiφqnen, πφ(δ) en= 1− q2n+2e n+1.
Note that,πφ(γβ) is a self-adjoint diagonal operator in the standard basis.
Remark 2.1 In this paper, we consider tensor products ofπ0. We could also consider
the representationπφ1 ⊗ πφ2, but this would not lead to more general results in this
paper, because representation labels only occur in phase factors; see [8, §II.A]. The
representation space of the tensor product representation is the Hilbert space
comple-tion of the algebraic tensor product of copies of 2(N).
Letσ : 2(N) ⊗ 2(N) → 2(N) ⊗ 2(N) be the flip operator, the linear operator
defined on pure tensors byσ(v1⊗ v2) = v2⊗ v1. We write
π12= (π0⊗ π0), π21= σ π12σ.
For three-fold tensor product representations, we write
π1(23)= (π0⊗ π0⊗ π0)(1 ⊗ )(), π(12)3= (π0⊗ π0⊗ π0)( ⊗ 1)().
Since is coassociative, we have π1(23)= π(12)3.
From (2.3), one finds
Using this, eigenvectors ofπ12(γ γ∗) can be computed (see [8] for details): for p∈ Z and x ∈ N define e12x,p= n,m∈N n−m=p Cx,m,nem⊗ en,
where we assume e−n = 0 for n ≥ 1, then π12(γ γ∗)ex12,p = q2xe12x,p. The Clebsch–
Gordan coefficients Cx,m,ncan be given explicitly in terms of Wall polynomials, see
[12], which are defined by
pn(qx; a; q) = 2ϕ1 q−n, 0 aq ; q, q x+1 = (−a)nq 1 2n(n+1) (aq; q)n 2ϕ0 q−n, q−x – ; q, qx a , (2.4)
for n, x ∈ N. The second expression follows from applying transformation [3, III.8]
with b → 0. Note that, for x ∈ N, the 2ϕ0-series can be considered as a
polyno-mial in q−nof degree x. This polynomial is (proportional to) an Al-Salam–Carlitz II
polynomial.
Let the function ¯pn(qx; a; q) be defined by
¯pn(qx; a; q) = (−1)n+x (aq)x−n(aq; q) ∞(aq; q)n (q; q)n(q; q)x pn(qx; a; q), (2.5)
then from the orthogonality relation for the Wall polynomials and from completeness, we obtain the orthogonality relations
x∈N ¯pn(qx; a; q) ¯pm(qx; a; q) = δnm, n∈N ¯pn(qx; a; q) ¯pn(qy; a; q) = δx y,
for 0 < a < q−1. The second relation corresponds to orthogonality relations for
Al-Salam–Carlitz II polynomials. The coefficients Cx,m,n, m, n ∈ N are defined by
Cx,m,n =
¯pn(q2x; q2(n−m); q2), n ≥ m,
¯pm(q2x; q2(m−n); q2), n ≤ m,
and they satisfy
Cx,n,m = Cx,m,n, (2.6)
which follows from the explicit expression as a2ϕ1-function. Furthermore, we define
The set{e12x,p | p ∈ Z, x ∈ N} is an orthonormal basis for 2(N) ⊗ 2(N). The
actions of theAq-generators on this basis are given by
π12(α) e12x,p = 1− q2xe12 x−1,p, π12(β) e12x,p = −q x+1 e12x,p+1, π12(γ ) e12x,p = q x e12x,p−1, π12(δ) e12x,p = 1− q2x+2e12 x+1,p, (2.7)
where e12−1,p= 0. We can also find eigenvectors e21x,pofπ21(γ γ∗) for eigenvalue q2x,
x∈ N: e21x,p= n,m∈N m−n=p Cx,m,nem⊗ en= e12x,−p.
3 6 j -symbols and q-Bessel functions
In [8], explicit expressions for the 6 j -symbols (and for more general coupling
coeffi-cients) have been found. It turns out that they are essentially q-Bessel functions. Here, we derive these results again using a more direct approach, and use this interpretation of the q-Bessel functions to obtain summation identities.
3.1 6 j -symbols
In the same way as above, we can find eigenvectors ofπ1(23)(γ γ∗) and π(12)3(γ γ∗);
for x ∈ N, p, r ∈ Z, e1x(23),p,r = n∈N Cx,n,n+pen⊗ e23n+p,x−n−r = n,m∈N Cx,n,n+pCn+p,m,ken⊗ em⊗ ek, n − m + k = x − r, e(12)3x,p,r = k∈N Cx,k−p,k ek12−p,r−x+k⊗ ek = k,m∈N Cx,k−p,kCk−p,n,men⊗ em⊗ ek, n − m + k = x − r,
are eigenvectors for eigenvalue q2x, x∈ N. We use here the convention e−n= e−n,p=
0 for n∈ −N≥1. The actions of theAq-generatorsα, β, γ, δ on the eigenvectors can
π1(23)(α)e1x(23),p,r = 1− q2xe1(23) x−1,p,r, π(12)3(α)e(12)3x,p,r = 1− q2xe(12)3 x−1,p,r, π1(23)(β)e1x(23),p,r = −qx+1e 1(23) x,p+1,r, π(12)3(β)e(12)3x,p,r = −qx+1e(12)3x,p+1,r, π1(23)(γ )e1x(23),p,r = qxe 1(23) x,p−1,r, π(12)3(γ )e(12)3x,p,r = qxe(12)3x,p−1,r π1(23)(δ)e1x(23),p,r = 1− q2x+2e1(23) x+1,p,r, π(12)3(δ)e(12)3x,p,r = 1− q2x+2e(12)3 x+1,p,r,
where e−1,p,r= 0. Note that, this corresponds exactly to the actions on the
eigenvec-tors ex,p.
The 6 j -symbol (or Racah coefficient) Rxp
1,r1;p2,r2 is the (re)coupling coefficient
between the two eigenvectors; Rxp 1,r1;p2,r2 = e1x(23),p1,r1, e(12)3x,p2,r2 , or equivalently e1x(23),p1,r1 = p2,r2 Rxp 1,r1;p2,r2e 1(23) x,p1,r1. (3.1)
We start by looking at some simple properties of R.
Proposition 3.1 The coefficients R have the following properties:
(i) Orthogonality relations:
p1,r1∈Z Rxp 1,r1;p2,r2R x p1,r1;p3,r3 = δp2,p3δr2,r3. (ii) Rxp 1,r1;p2,r2 = R x p1+k,r1;p2+k,r2 for k∈ Z. (iii) Rxp 1,r1;p2,r2 = R x+k p1,r1;p2,r2for k∈ Z≥−x. (iv) For k, m, n ∈ N, Cx,n+p1,nCn+p1,m,k = p2∈Z≤k Rxp1,r;p2,rCx,k−p2,kCk−p2,m,n, x− r = n − m + k. (v) Duality: Rxp 1,r;p2,r = R x −p2,r;−p1,r.
Note that, identity (iii) implies that R is independent of x; therefore, we will omit the superscript ‘x.’
Proof The coefficients R are matrix coefficients of a unitary operator, which leads to
the orthogonality relations. The next two identities follow from the∗-structure of Aq.
Fromβ∗= −qγ , we obtain e1x(23),p 1±1,r1, e (12)3 x,p2,r2 =e1x(23),p1,r1, e(12)3x,p 2∓1,r2 ,
which implies (ii). Identity (iii) follows fromα∗ = δ. Identity (iv) follows from the
expansion
e1x(23),p1,r1 =
p2,r2∈Z
by taking inner products with en⊗em⊗ek. The duality property follows from identity (iv). 3.2 q-Bessel functions Define Jν(x; q) = xν2(q ν+1; q)∞ (q; q)∞ 1ϕ1 0 qν+1; q, qx , x≥ 0, ν ∈ R, (3.2)
which is Jackson’s third q-Bessel function (also known as the Hahn-Exton q-Bessel
function), see e.g., [16]. Note that,
(B; q)∞ 1ϕ1 A B; q, Z =∞ k=0 (A; q)k(Bqk; q)∞ (q; q)k (−1) k q12k(k−1)Zk
is an entire function in B, so we may takeν to be a negative integer in (3.2); in this
case, we have the identity
J−n(x; q) = (−1)nqn2Jn(xqn; q), n ∈ N,
see [16, (2.6)]. We will use the following generating function to identify the 6 j -symbols
with q-Bessel functions. Proposition 3.2 For|t| < 1, ∞ m=0 q−νm2 Jν(xqm; q) t m (q; q)m = x ν 2(q ν+1; q)∞ (q, t; q)∞ 1ϕ1 t qν+1; q, qx .
Proof Write Jνas a1ϕ1-series, interchange the order of summation, and use
summa-tion formula [3, (II.1)];
∞ m=0 1ϕ1 0 qν+1; q, xq m+1 tm (q; q)m = ∞ k=0 (−1)kq12k(k−1)(xq)k (q, qν+1; q)k ∞ m=0 qmktm (q; q)m =∞ k=0 (−1)kq12k(k−1)(xq)k (q, qν+1; q)k(tqk; q) ∞.
If t−1qν+1∈ q−N, the right-hand side in the Proposition3.2can be written in terms
Corollary 3.3 For n∈ N, ∞ m=0 q−12(ν−n)mJν−n(xqm; q)q m(ν+1) (q; q)m = x 1 2(ν−n)(qx; q)∞ (q; q)∞ pn(q ν; x; q).
Proof In Proposition3.2replaceν by ν − n, set t = qν+1, and use the transformation
1ϕ1 A B; q, Z = (A, Z; q)(B; q) ∞ ∞ 2ϕ1 0, B/A Z ; q, A ,
(which is a special case of [3, (III.4)]) and the Definition (2.4) of the Wall polynomials.
We are now in a position to show that the 6 j -symbols are essentially q-Bessel functions. Proposition 3.4 For p1, p2, r1, r2∈ Z, Rp1,r1;p2,r2 = δr1,r2(−q) p1−p2J r1(q 2 p1−2p2; q2).
Proof We write out Proposition3.1(iv) for m= k = 0, and we replace p2by−p2,
p2∈N Rp1,r;−p2,r (−1)p2qp2(n+x+1) (q2; q2) p2 =(−1)(qp12q; qp12(x−n+1)) p1 pn(q2x; q2 p1; q2), x− r = n,
then the result follows from Corollary3.3.
3.3 Identities
Several classical identities for 6 j -symbols for SU(2) remain valid for our 6 j-symbols.
By Proposition3.4, these can be interpreted as identities for q-Bessel functions.
First of all, the orthogonality relations for the 6 j -symbols from Proposition3.1are
equivalent to the well-known q-Hankel orthogonality relations, see [16, (2.11)], for
the q-Bessel functions Jν.
Theorem 3.5 For n, m ∈ Z,
x∈Z
To derive other identities, it is convenient to represent eigenvectors ofγ γ∗as binary
trees; see e.g., Van der Jeugt’s lecture notes [20] for more details. We denote
e12x,n2−n1 =
x
n1 n2
where n1, n2, x ∈ Z. Equivalently, we can identify this tree with the Clebsch–Gordan
coefficient Cx,n1,n2, similar as in [18]. The identity e
12 x,p= e21x,−p, which is equivalent to (2.6), is represented as x n1 n2 = x n2 n1 (3.3)
where p= n1− n2. By coupling two of these, we can represent eigenvectors
corre-sponding to threefold tensor products:
e1x(23),p1,r123 = x n1 n2 n3 p 1 e(12)3 x,p2,r123 = x n1 n2 n3 p 2
where p 1= n1+ p1, p 2= n3− p2, and ri j k = x −ni+nj−nkfor i, j, k ∈ {1, 2, 3}.
Now we can e.g., represent the identities e1x(23),p1,r123 = e
1(32) x,p1,r132 = e (23)1 x,−p1,r231 by x n1 n2 n3 p 1 = x n1 n3 n2 p 1 = x n3 n2 n1 p1
The transition (3.1) from e1x(23),p1,r to e
(12)3
x,p2,r which involves a 6 j -symbol, which is
equivalent to identity (ii) in Proposition3.1in terms of Clebsch–Gordan coefficients,
is represented as x n1 n2 n3 p 1 R x,n1,n2,n3 p1 ,p2 x n1 n2 n3 p 2
where the coefficient R is given by Rx,n1,n2,n3 p1 ,p2 = Rp1,r123;p2,r123 = (−q)p 1+p 2−n1−n3J x−n1+n2−n3(q 2 p 1+2p 2−2n1−2n3; q2). (3.4)
Note that the transition from right to left involves exactly the same 6 j -symbol. To find identities for the 6 j -symbols, we can use the binary trees and identities for these trees as explained above, without referring to the underlying eigenvectors. We obtain the following identities, which can be considered as analogs of Racah’s backcoupling identity, the Biedenharn–Elliot (or pentagon) identity, and the hexagon identity. Theorem 3.6 The following identities hold:
(i) Rx,n1,n2,n3 p1,p2 = p∈Z Rx,n1,n3,n2 p1,p R x,n3,n1,n2 p,p2 ,
or in terms of q-Bessel functions Jr123(q p1+p2; q) = p∈Z Jr132(q p+p1; q)J r312(q p+p2; q)qp, where ri j k = x − ni + nj− nk. (ii) Rx,n1,n2,p1 r1,p2 R x,p2,n3,n4 p1,r2 = p∈Z Rr1,n2,n3,n4 p1,p R x,n1,p,n4 r1,r2 R r2,n1,n2,n3 p,p2 ,
which in terms of q-Bessel functions is equivalent to the product formula Jν+μ1(q P−Q; q)J ν+μ2(q Q−R; q) = μ∈Z Aμ1,μ2,μ P,Q,R Jν+μ(q P−R; q) where P, Q, R, ν, μ1, μ2∈ Z and Aμ1,μ2,μ P,Q,R = (−1)μ1+μ2qμ− 1 2(μ1+μ2) × Jμ2−μ1+P−Q(qμ−μ1; q)Jμ1−μ2+Q−R(qμ−μ2; q). (iii) r∈Z Rx,p1,n3,n4 p2,r R r,n2,n1,n3 p3,p1 R x,p3,n2,n4 p4,r = r∈Z Rx,n1,n2,p2 r,p1 R r,n2,n4,n3 p2,p4 R x,n1,n3,p4 r,p3 ,
r∈Z (−1)p2+p4qr−n4+12(p2+p4)J r−n2+n1−n3(q p1+p3−n2−n3; q) × Jx−p1+n3−n4(q r+p2−p1−n4; q)J x−p3+n2−n4(q r+p4−p3−n4; q) = idem(n1, n2, p1, p3) ↔ (n4, n3, p2, p4) .
Here ‘idem’ means that the same expression is inserted but with the parameters interchanged as indicated.
Proof The first identity follows from
x n1 n2 n3 p1 x n1 n2 n3 p2 x n1 n3 n2 p Rx,n1,n2,n3 p1,p2 Rx,n1,n3,n2 p1,p R x,n3,n1,n2 p,p2
The second identity is
x n1 n2 n3 n4 r1 p1 x n1 n2 n3 n4 p2 p1 x n1 n2 n3 n4 p2 r2 x n1 n2 n3 n4 r1 p x n1 n2 n3 n4 r2 p Rx,n1,n2,p1 r1,p2 R x,p2,n3,n4 p1,r2 Rrp11,n,p2,n3,n4 Rxr1,n,r12,p,n4 Rrp2,p,n21,n2,n3
The corresponding identity for q-Bessel functions is obtained by substituting
r1− n1= P, p1− p2= Q, n4− r2= R, ν = x − n1,
The third identity is x n1 n2 n3 n4 p1 p2 x n1 n2 n3 n4 p1 r x n1 n3 n2 n4 p3 r x n1 n3 n2 n4 p3 p4 x n1 n2 n3 n4 p2 r x n1 n3 n2 n4 p4 r Rx,p1,n3,n4 p2,r R r,n2,n1,n3 p3,p1 Rx,p3,n2,n4 p4,r Rx,n1,n2,p2 r,p1 Rr,n2,n4,n3 p2,p4 Rx,n1,n3,p4 r,p3 Remark 3.7 (i) The q-Hankel transform of a function f ∈ L2(qZ; qx) is defined by
(Hνf)(n) =
x∈Z
f(qx)Jν(qx+n; q)qx, n∈ Z.
Identity (i) of Theorem3.6shows that the q-Hankel transform maps an
orthogo-nal basis of q-Bessel functions to another orthogoorthogo-nal basis of q-Bessel functions,
which implies a factorization of the q-Hankel transform: Hr123 = Hr312Hr132.
(ii) Identity (ii), the product formula for q-Bessel functions, has appeared before
in the literature; representation theoretic proofs are given by Koelink in [13,
Corollary 6.5] and Kalnins et al. in [11, (3.20)]. A direct analytic proof is given
by Koelink and Swarttouw in [14].
(iii) It is well known that the hexagon identity for classical 6 j -symbols can be interpreted as a quantum Yang–Baxter equation. Here, we obtain an
infinite-dimensional solution: for u, v ∈ Z, define a unitary operator R(u, v) : 2(Z) ⊗
2(Z) → 2(Z) ⊗ 2(Z) by
R(u, v)(ex−a⊗ eb−x) =
y∈Z
Rxu,a,v,b,y eb−y⊗ ey−a, a, b, x ∈ Z,
where{ex | x ∈ Z} is the standard orthonormal basis for 2(Z). Then the hexagon
identity says that the operatorR satisfies
R12(u, w)R13(v, w)R23(u, v) = R23(u, v)R13(v, w)R12(u, w)
as an operator identity on 2(Z) ⊗ 2(Z) ⊗ 2(Z).
4 3n j -symbols and multivariate q-Bessel functions
We consider certain 3n j -symbols and show that these can be considered as mul-tivariate q-Bessel functions, which are limits of the mulmul-tivariate Askey–Wilson
polynomials introduced by Gasper and Rahman in [4]. In this section, we use the
following notation. For v = (v1, v2, . . . , vd−1, vd), we define |v| = dj=1vj and
ˆv = (vd, vd−1, . . . , v2, v1). For some function f : Zd → C, we set
x f(x) = xd∈Z · · · x1∈Z f(x1, . . . , xd),
provided the sum converges.
4.1 Multivariate q-Bessel functions
Let d ∈ N≥1. Forν = (ν0, . . . , νd+1) ∈ Zd+2, we define q-Bessel functions in the
variables x = (x1, . . . , xd), λ = (λ1, . . . , λd) ∈ Zdby Jν(x, λ) = d j=1 Jνj−xj+1−λj−1(qxj−xj+1+λj−λj−1; q), (4.1)
whereλ0= ν0and xd+1= νd+1. Occasionally, we will use the notation Jν(x, λ; q)
to stress the dependence on q.
Theorem 4.1 The multivariate q-Bessel functions have the following properties: (i) Orthogonality relations:
x
Jν(x, λ)Jν(x, λ )qx1 = δ
λ,λ qνd+1+ν0−λd, λ, λ ∈ Zd.
(ii) Self-duality: Jν(x, λ) = Jˆν(ˆλ, ˆx).
Proof The self-duality property follows directly from (4.1). The orthogonality rela-tions follow by induction using the q-Hankel orthogonality relarela-tions from Theorem
3.5, which can be written as
xj∈Z Jνj−xj+1−λj−1(qxj−xj+1+λj−λj−1; q)Jνj−xj+1−λj−1(q xj−xj+1+λ j−λj−1; q)qxj = δλj,λ jq xj+1−λj+λj−1. (4.2) Define for k = 1, . . . , d + 1, Jν(k)(x, λ) = d j=k Jνj−xj+1−λj−1(q 1 2(xj−xj+1+λj−λj−1); q),
the empty product being equal to 1. Note that Jν(1)= Jν and
We will show that xk∈Z · · · x1∈Z Jν(x, λ)Jν(x, λ )qx1 (4.4) = δλ1,λ 1· · · δλk,λ kq xk+1−λk+λ0 J(k+1) ν (x, λ)Jν(k+1)(x, λ ). (4.5)
For k= 1, (4.4) follows directly from (4.2). Now assume that (4.4) holds for a certain
k, then by (4.2) and (4.3), xk+1∈Z · · · x1∈Z Jν(x, λ)Jν(x, λ )qx1 = δλ1,λ 1· · · δλk,λ k xk+1∈Z Jν(k+1)(x, λ)Jν(k+1)(x, λ )qxk+1−λk+λ0 = δλ1,λ 1· · · δλk+1,λ k+1J (k+2) ν (x, λ)Jν(k+2)(x, λ )qxk+2−λk+1+λ0,
which proves the orthogonality relations.
Next we show that the multivariate q-Bessel functions can be considered as limit cases of multivariate Askey–Wilson polynomials. The 1-variable Askey–Wilson poly-nomials are defined by
pn(x; a, b, c, d | q) = (ab, ac, ad; q) n
an 4ϕ3
q−n, abcdqn−1, ax, a/x
ab, ac, ad ; q, q
,
which are polynomials in x+x−1of degree n, and they are symmetric in the parameters
a, b, c, d. Using notation as in [9], the multivariate Askey–Wilson polynomials are
defined as follows. Let n= (n1, . . . , nd) ∈ Ndand x = (x1, . . . , xd) ∈ (C×)d, then
the d-variable Askey–Wilson polynomials are defined by
Pd(n; x; α | q) = d j=1 pnj xj; αjqNj−1,α j α2 0 qNj−1,αj+1 αj xj+1,α j+1 αj x−1j+1| q , (4.6) where Nj = j k=1nk, N0 = 0, α = (α0, . . . , αd+2) ∈ C d+3, x d+1= αd+2. These
are polynomials in the variables x1+ x1−1, . . . , xd+ xd−1of degree|n| = Nd.
Proposition 4.2 Letλ = (λ1, . . . , λd) ∈ Zd,ν = (ν0, . . . , νd+1) ∈ Zd+2and define
α(m) =q−m, q12ν0, q 1 2ν1−m, . . . , q 1 2νj− jm, . . . , q 1 2νd−dm, qνd+1+m ∈Cd+3, x(m) = q−12ν0−x1+m, q 1 2ν1−ν0−x2+m, . . . , q 1 2νd−1−ν0−xd+m ∈ Cd, λ + m = (λ1+ m, . . . , λd+ m) ∈ Nd, Cm(x; λ; α) = d j=1 q(12νj−1−νj+ν0+xj+1−m)(λj+m) qνj−νj−1−2m; q λj+m ,
then lim m→∞ Pd(λ + m; x(m); α(m) | q) Cm(x; λ; α) = (q; q)d ∞ ⎛ ⎝ d j=1 q12(xj+1−xj+j+1−j)(νj−xj+1−j−1) ⎞ ⎠ Jν(x, ), where = (1, . . . , d) with j = ν0− j k=1λkand0= ν0.
Proof First we substitute
α0→ q−m, nj → λj+ m, j = 1, . . . , d
xj → xjqm, αj → αjqm( j−1), j = 1, . . . , d + 1,
in (4.6) (recall, xd+1= αd+2). The4ϕ3-part of the j th factor pnj is
4ϕ3 ⎛ ⎜ ⎝ q−λj−m, α2 j+1q 2( kj−1=1λk)+λj−1+m,αj+1 αj xj+1xjq m,αj+1xj+1 αjxj q −m α2 j+1 α2 j q−2m, αj+1xj+1q2m+ j−1 k=1λk, α j+1xj+1q j−1 k=1λk ; q, q ⎞ ⎟ ⎠,
where the empty sum equals 0. Letting m→ ∞, this function tends to
1ϕ1 0 αj+1xj+1q j−1 k=1λk ; q, αjxj+1 αj+1xj q1−λj . Finally, we substitute αj → q 1 2νj−1, xj → q21νj−1−ν0−xj, and setν0− j
k=1λk = j for j= 0, . . . , d, then we have
1ϕ1 0 qνj−xj+1−j−1 ; q, q xj+1−xj+j−j−1 ,
which we recognize as the 1ϕ1-part of the j th factor of the multivariate q-Bessel
function Jν(x, ), see (4.1).
4.2 3n j -symbols
Let k ∈ N≥1, and let r, s ∈ Zk, n∈ Zk+2. We define the 3n j -symbols Rrx,s;nto be the
coupling coefficients between two specific binary trees corresponding to(k + 2)-fold
x n1 n2 nk+1 nk+2 r1 rk · · · = x r n x n1 n2 nk+1 nk+2 s1 sk · · · = x s n
Note that, a node with a bold symbol represents several nodes, and that the label r (respectively s) on the right (left) of a node means that all branches ‘hang’ on the right
(left) edge. The 3n j -symbols Rxr,s,nare defined by
x r n = s Rrx,s,n x s n
and we will denote the corresponding transition again by an arrow. Note that, for k= 1
we have Rrx,s,n= Rrx,s,n1,n2,n3.
Proposition 4.3 The coefficients Rrx,s,nhave the following properties:
(i) Orthogonality relations:
rR x,n r,s R x,n r,s = δs,s (ii) Duality: Rxr,s,n = Rˆs,ˆrx,ˆn.
Proof The coefficients R are the matrix coefficients of a unitary operator, which implies the orthogonality relations. The duality property is a consequence of the iden-tity x r n = x ˆr ˆn
which follows from repeated application of (3.3).
Theorem 4.4 For i= 1, 2 let ki ∈ N≥1, ni ∈ Zki+1and ri, si ∈ Zki. Let k= k1+k2,
n= (n1, n2), r = (r1, r2), s = (s1, s2), then Rrx,s,n= R x,(n1,rk1+1) r1,s1 R x,(sk1,n2) r2,s2 .
As a consequence, Rrx,s,n= k j=1 Rrxj,s,sjj−1,nj+1,rj+1, where s0= n1and rk+1= nk+2.
Proof The first identity follows from
x r2 r1 n1 n2 Rrx,(n1,s11,rk1+1) x r2 s1 n1 n2 Rrx,(s2,sk12 ,n2) x s1 s2 n1 n2
The second identity follows from repeated application of the first identity.
From (3.4) it follows that Rrx,s,nis essentially a multivariate q-Bessel function as defined
by (4.1).
Corollary 4.5 Letν(x, n) = (n1, x + n2, . . . , x + nk+1, nk+2), then
Rrx,s,n= (−q)r1+sk−n1−nk+2Jν(x,n)(r, s; q2).
Note that, this corollary and Proposition4.3together give a representation theoretic
proof of Theorem4.1.
Our next goal is to prove a summation identity for the multivariate q-Bessel functions. Let us first mention that by interpreting a binary tree as a product of Clebsch–
Gordan coefficients, the 3n j -symbols Rrx,s,nsatisfy, by definition, the formula
Cx,r,n= s Rrx,s,nCx,ˆs,ˆn, Cx,r,n= k +1 j=1 Crj−1,nj,rj, (4.7)
where r0= x, rk+1= nk+2, s0= n1, sk+2= x. The functions Cx,r,ncan be
consid-ered as multivariate Wall polynomials, which are q-analogs of Laguerre polynomials.
In this light, (4.7) is a multivariate q-analog of an identity proved by Erdélyi [2] which
states that the Hankel transform maps a product of two Laguerre polynomials to a product of two Laguerre polynomials.
For the 3n j -symbols Rrx,s,n, there exists a multivariate analog of the Biedenharn–
Elliott identity. In terms of q-Bessel functions, this gives an expansion formula for k-variable q-Bessel functions in terms of (k − 1)-variable q-Bessel functions. The
identity requires also another 3n j -symbol. For r, s ∈ Zk, n∈ Zk+2, x ∈ Z, let Srx,s,n
x r n = ˆs Srx,s,n x s n (4.7) Note that, ˆs = s 1· · ·
sk. This 3n j -symbol can of course also be considered as
a multivariate q-Bessel function (see the following result), but it lacks the self-duality property. Let us first express S in terms of the 6 j -symbols.
Lemma 4.6 Srx,s,nis given by Srx,s,n= k j=1 Rrsjj+1,sj,n1,rj−1,nj+2, with sk+1= x and r0= n2.
Proof We use the transition sk− j+1 rk− j n1 rj nk− j+2 Rrskk− j+1− j,sk− j,n1,rk− j−1,nk− j+2 sk− j+1 sk− j n1 rj nk− j+2 where r j = rk− j−1 r j nj
and where r j = (r1, . . . , rk− j−2) and nj = (n2, . . . , nk− j+1). We set sk+1= x and
r0= n2, then applying this transition successively on subtrees for j = 0, . . . , k − 1
gives Srx,s,n = k−1 j=0 Rrskk− j− j+1,sk,n− j1,rk− j−1,nk− j+2.
Changing the index gives the stated expression for the coupling coefficient S.
The following identity is the multivariate analog of the Biedenharn–Elliott identity
from Theorem3.6, i.e., the k= 2 case gives back Theorem3.6(ii).
Theorem 4.7 For k∈ N≥2let r, s ∈ Zkand n∈ Zk+2, then Rxr,s,n = t∈Zk−1 S(t,rx,n 1),sR r1,n r ,t ,
where v is obtained from v by leaving out the first component. In terms of multivariate q-Bessel functions,
Jν(x,n)(r, s) =
t∈Zk−1
Ar1
with Ar1 t,s= (−q 1 2)|t|+|s|−|n|−(k−2)n1−sk+r2 × k j=1 Jsj+1−n1+tj−1+nj+2(qsj+tj−n1−nj+2; q), tk= r1.
Proof This follows from the transition
x r n = x r r1 n1 n Rrr1 ,t,n x r r1 t n1 n = x p n Spx,s,n x s n
where p= (t, r1), and the definition of the coupling coefficients R.
Remark 4.8 It seems that there are no analogs for the 3n j -symbols R of identities (i)
and (iii) of Theorem3.6, but there does exist an analog of Theorem3.6(i) involving
only the 3n j -symbols S which may be of interest. This is obtained as follows.
Let n ∈ Zk+2. For j ∈ {1, 2, . . . , k + 1}, we define n
j = (nk+3− j,
. . . , nk+2, n1, . . . , nk+2− j). Furthermore, given a vector v, we denote (as in
Theo-rem 4.7) by v the vector v without the first component, and we set n j = (nj) .
Consider the transition
x ˆr ˆn = x r nk+2 n 1 Sx,n1 r,s1 x s1 nk+2 n 1 = x ˆs1 n1
Iterating this transition k+1 times shows that the coupling coefficient in the transition
x ˆr ˆn Trx,n,s x ˆs nk+1 = x s n1 n is given by Trx,s,n=· · ·Sx,n1 s0,s1· · · S x,nk+1 sk,sk+1 , s0= r, sk+1= s.
On the other hand, by the definition of the coupling coefficient S, we have x ˆr ˆn = x r n Ssx,n,r x s n = x s n1 n . so that Ssx,r,n= sk · · · s1 Sx,n1 s0,s1· · · S x,nk+1 sk,sk+1 , s0= r, sk+1= s.
For k= 1, this gives back Theorem3.6(i).
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